Portraits of Wildflowers

Perspectives on Nature Photography

Aztec dancer and ant

with 71 comments

Aztec Dancer Damselfly with Ant 9632

I took this photograph close to a waterfall off Harrogate Dr. in northwest Austin last year on 7/24. Whether the ant ran any risk of getting eaten by the Aztec dancer damselfly, Argia nahuana, I don’t know, but the date reminds me of something I do know, namely that 7 and 24 are the perpendicular sides of a 7-24-25 right triangle because 7 squared plus 24 squared equals 25 squared. Other right triangles with the shortest side an odd number are  5-12-13,  9-40-41,  11-60-61,  13-84-85,  and the familiar 3-4-5. Can you figure out how to get the two longer sides of each right triangle of this type if you know only the shortest side?

(Speaking of math, did anyone notice that the number 63 that played a role in yesterday’s post can be written in base 2 as 111111?)

© 2015 Steven Schwartzman


Written by Steve Schwartzman

July 27, 2015 at 5:32 AM

71 Responses

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  1. Great capture! What a beautiful animal…


    July 27, 2015 at 5:35 AM

  2. ….it is truly gorgeous. as for the 7/24, there is help available, steve, if you’ll just reach out and call…..


    July 27, 2015 at 5:45 AM

    • I tried calling the damselfly on 7/24, Lance. It wouldn’t come to me so I had to slowly go to it.

      Steve Schwartzman

      July 27, 2015 at 5:56 AM

  3. I learn something everyday!


    July 27, 2015 at 6:02 AM

  4. No, I didn’t notice and honestly couldn’t follow the lesson. Too old to learn new math. OTOH, I could follow your nice parallel capture showing off all those nice hairs and wing venation. I can’t see them at this resolution, but I bet at full there are plenty of eye facets to enjoy.

    Steve Gingold

    July 27, 2015 at 6:19 AM

    • I took a look at a larger version of the image, and although the nearer eye appears to be in focus, I could see some curving rows of faint facets only in the upper part. I’ve captured clearly delineated facets in other photographs of damselflies and dragonflies over the years, but not in this one.

      No one is ever too old to learn new math!

      Steve Schwartzman

      July 27, 2015 at 6:32 AM

  5. Show off. Love your photos. Missy McIver friend of Ariana and cousin of her husband Michael

    Ada McIver

    July 27, 2015 at 6:29 AM

    • Welcome, Missy. When it comes to damselflies and mathematics, there’s every reason to show off, wouldn’t you say?

      Steve Schwartzman

      July 27, 2015 at 6:41 AM

  6. No, no, no and another no for good measure. Although I suppose if I sat quietly for long enough I might change the no to yes. In the meantime, I will simply enjoy the damselfly and the ant.


    July 27, 2015 at 6:59 AM

    • Hey, that quadruple no is no way to end your day, so I’m glad to see the negatives tempered into a possible positive in your second sentence. And there’s also the fact that because in multiplication a double negative makes a positive, your four no’s amount to a yes.

      Steve Schwartzman

      July 27, 2015 at 7:07 AM

  7. That is one gorgeous color dragonfly – it’s too early in the morning for math 🙂


    July 27, 2015 at 7:30 AM

    • It’s never too early in the morning for math in my brain, Nora, but no one will fault you for wanting to focus on the damselfly instead.

      Steve Schwartzman

      July 27, 2015 at 7:41 AM

  8. I’m frustrated, because trying to use the ALT code for the squared sign keeps throwing me off your page. Who knows what that’s about.

    Anyway — two cups of coffee and a good chunk of time have reminded me of the Pythagorean theorem, the little formula for consecutive integers, and a hint of how to do it. Pardon the verbal substitutions, but I don’t have time to solve that problem. Somehow, it involves x(squared), (x+2)(squared) and (x+4) squared. I think I’ve figured out how to solve for x, if x is the shortest side, but I can’t go the other direction. I’m just tickled I got this far.

    Now, off to see if I can find a damselfly as lovely as yours at work today.


    July 27, 2015 at 7:35 AM

    • I don’t think there’s a way in WordPress to do superscripts, and that’s why I wrote out the word squared, as you did. Because of the Pythagorean Theorem, squaring is indeed involved in finding a way to go from the shortest side to the other two sides, but I’ll say no more for now so you and (I hope) other people can think about it. Using x, x + 2, and x + 4 isn’t my approach to the problem, but there are often multiple ways to solve a math problem, and you might succeed with your approach.

      I’ve taken plenty of photographs of damselflies and dragonflies over the years, but I don’t recall ever having an ant in one. Ants typically move quickly, and there wasn’t one in the previous picture that I took 20 seconds earlier, nor in the following one that I took 20 seconds later. In looking back just now through all the 21 pictures I took of the damselfly, I found this is the only one that has an ant in it.

      Steve Schwartzman

      July 27, 2015 at 8:42 AM

  9. Beautiful image.

    I haven’t had enough coffee to contemplate the math. Maybe later 😀

    Sammy D.

    July 27, 2015 at 8:27 AM

    • Your comment and the previous one make me think that maybe our math classrooms should be equipped with coffee dispensers. It’s worth an experiment to see if math scores go up.

      Steve Schwartzman

      July 27, 2015 at 8:45 AM

  10. 😀 Thanks for the lovely image and strain on my brain and all the hilarious comments! 🙂


    July 27, 2015 at 9:25 AM

    • I like your rhyming phrase “strain on my brain.” I see that the math didn’t completely drain your brain, or else you wouldn’t have been able to leave your comment.

      Steve Schwartzman

      July 27, 2015 at 9:34 AM

  11. I probably could, Steve, but I’d much rather gaze at that wonderful blue Aztec dancer 🙂


    July 27, 2015 at 9:40 AM

  12. Haha! It’s too early for math. I’m curious to know…I’ll put the problem to my math-minded husband, who is currently studying differential geometry for FUN.

    But it’s never too early for a beautiful damselfly. I got tired of waiting for others’ insect posts, so I made my own, though my macros are nowhere near yours in clarity. Most were taken with the older Olympus, but a few with the Canon. (http://wp.me/p28k6D-1OE)

    Thanks for throwing me a bone, er, uh exoskeleton!


    July 27, 2015 at 10:35 AM

    • Your reaction to the math continues in the vein of almost all the previous ones. Do let us know if your husband figures out the right triangles. I’ll probably post a solution in a few days.

      I’m glad to see you took the bit between your teeth and ran with it, insect-wise, that is. I’ll wish us all exhilarating exemplary exoskeletons.

      Steve Schwartzman

      July 27, 2015 at 11:02 AM

  13. Where x is the shortest side, y is the longer side, z is the hypotenuse,
    y = ((x-1)/2)*(x+1)
    z = y+1
    And 63 also = hex ff.
    I miss math and (sorry if you’re a pure math person) physics.


    July 27, 2015 at 10:48 AM

    • Hooray, a solution to the right triangle problem! Were you already aware of that method of generating right triangles from odd numbers, or did you figure it out just now? I can’t remember how I first learned it, but for decades I showed it to my high school math students. Last year for some reason I started playing around and figured out a variant of that method that works when the shortest side is a multiple of 4; it produces 8-15-17, 12-35-37, 16-63-65, 20-99-101, etc., where the two longer sides end up being consecutive odd integers rather than just consecutive integers.

      In your hexadecimal conversion there’s a typo: the hex equivalent of 63 is 3F. FF is 1 less than the third power of 16, which is to say 255. Do you want me to change it in your comment and then erase this part of my reply?

      You’re correct that I incline more toward math for math’s sake rather than toward its embodiment in the physical world. I miss teaching it.

      Steve Schwartzman

      July 27, 2015 at 11:27 AM

      • Don’t bother changing the hex conversion error. I don’t mind making a silly mistake. I looked it up in a table, but read the wrong column. Of course, if I’d estimated, I’d have known. In fact, I did have that intuition, but ignored it.

        I’d never heard of this series, and just worked out the equation for fun.

        My exposure to math comes through physics and engineering. I love how strange fields of mathematics prove so useful at describing the physical world, and. thus, that reality is stranger than it seems. Lately I am developing some fascination with math itself, particularly ancient mathematical mysticism.


        July 28, 2015 at 5:12 AM

        • I taught math on and off for decades, and I used to pursue mathematical relationships for the fun of it to see what I might find. I’m always amazed when I find that the same structure underlies two unrelated things, for example the multiplication of negatives and positives and also the addition of odds and evens.

          I’m aware that some (probably most) physicists take a very mathematical view of the physical world, with the extreme statement of that view being that everything is math (an idea I’ve toyed with myself since I was in college).

          Steve Schwartzman

          July 28, 2015 at 7:55 AM

          • Ok, I’ll tell you a true story. Years ago, a boyfriend and I were discussing how to best set up some rules for my children, a very human topic. In the middle of the discussion, he looked at me and asked, “Do you ever have difficulty translating equations into words?” I asked him if he was referring to our current topic, and he said yes. I assured him that I did not, but have always wondered what he was talking about.


            July 29, 2015 at 6:56 PM

            • Why didn’t you ask him for an explanation at the time? Is it too late to ask him now?

              Math students usually have more trouble going the other way, i.e. taking a description of something and creating an equation (or equations) to represent the situation.

              Steve Schwartzman

              July 29, 2015 at 7:15 PM

              • I don’t know where he is anymore. He said his college advisors tried to get him to do a math PhD, but he didn’t think he had enough talent. He clued me in how interesting mathematical minds are.


                July 30, 2015 at 6:31 AM

                • The Internet is pretty good at finding people, but you may prefer to let boyfriends be bygones.

                  Steve Schwartzman

                  July 30, 2015 at 6:45 AM

              • This is cool. http://www.sacred-texts.com/eso/sta/sta16.htm says that it’s a paraphrase of Thomas Taylor’s Theoretic Arithmetic. Here’s a snippet:
                “Even numbers are also divided into three other classes: superperfect, deficient, and perfect.

                Superperfect or superabundant numbers are such as have the sum of their fractional parts greater than themselves. For example: 1/2 of 24 = 12; 1/4 = 6; 1/3 = 8; 1/6 = 4; 1/12 = 2; and 1/24 = 1. The sum of these parts (12+6+8+4+2+1) is 33, which is in excess of 24, the original number.

                Deficient numbers are such as have the sum of their fractional parts less than themselves. For example: 1/2 of 14 = 7; 1/7 = 2; and 1/14 = 1. The sum of these parts (7+2+1) is 10, which is less than 14, the original number.

                Perfect numbers are such as have the sum of their fractional parts equal to themselves. For example: 1/2 of 28 = 14; 1/4 = 7; 1/7 = 4; 1/14 = 2; and 1/28 = 1. The sum of these parts (14+7+4+2+1) is equal to 28.”
                “The Pythagoreans evolved their philosophy from the science of numbers. The following quotation from Theoretic Arithmetic is an excellent example of this practice:

                “Perfect numbers, therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be. And evil indeed is opposed to evil, but both are opposed to one good. Good, however, is never opposed to good, but to two evils at one and the same time. Thus timidity is opposed to audacity, to both [of] which the want of true courage is common; but both timidity and audacity are opposed to fortitude. Craft also is opposed to fatuity, to both [of] which the want of intellect is common; and both these are opposed to prudence. Thus, too, profusion is opposed to avarice, to both [of] which illiberality is common; and both these are opposed to liberality. And in a similar manner in the other virtues; by all [of] which it is evident that perfect numbers have a great similitude to the virtues. But they also resemble the virtues on another account; for they are rarely found, as being few, and they are generated in a very constant order. On the contrary, an infinite multitude of superabundant and diminished numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite.””
                My hermetic teacher tells me that Pythagoras had a great mystical school, but all the members held their secrets so well that the teachings died with them. This teacher also tells me that the goal is to be neither good nor evil, but to be in balance between the two, and this is just what the above quote says. Thanks, Steve. Hope you enjoy it also.


                July 30, 2015 at 6:54 AM

                • Thanks. So your intuition was correct about analogies of perfect numbers to human experience, and as expressed in your quotation to virtues in particular. I’ve never seen such a detailed explanation of the analogy before. (By the way, the books I’ve read have used the term abundant rather than superabundant or superperfect.)

                  There are also pairs of numbers called amicable, in which the proper divisors of one of them add up to the other, and vice versa:


                  Supposedly a boyfriend-girlfriend or husband-wife would each wear a locket that bore one of the numbers in an amicable pair, e.g 220 and 284.

                  Steve Schwartzman

                  July 30, 2015 at 7:11 AM

                • Thank you for the amicable number info. I can imagine how much more mystical number must have seemed to the ancients, where only an elite few had the opportunity to become familiar with the concepts.


                  July 30, 2015 at 7:28 AM

                • Based on the reports I keep reading about low math tests scores and the widespread ignorance and fear of mathematics in the United States, I’m tempted to say that only an elite few are familiar with arithmetic here as well!

                  Steve Schwartzman

                  July 30, 2015 at 7:46 AM

                • Amen.


                  July 30, 2015 at 9:01 AM

        • Speaking of mathematical mysticism, I just noticed that today is July 28, and the ancient Greek mathematicians considered 28 a “perfect” number because it equals the sum of all its lesser (positive) divisors: 28 = 1 + 2 + 4 + 7 + 14.

          Steve Schwartzman

          July 28, 2015 at 7:58 AM

          • Don’t you wonder if they had analogies of perfect numbers to human experience, some kind of “as above so below”?


            July 29, 2015 at 6:58 PM

            • That’s an interesting and plausible hypothesis. I don’t recall if I ever read an account of how/why the category of perfect numbers was created, but historians of mathematics may know.

              Steve Schwartzman

              July 29, 2015 at 7:09 PM

    • I found the pattern but the equation form above is better. Here is what I saw.
      3*1+1= the longer side

      The relationship between the two sides is then easy to see.


      July 27, 2015 at 8:52 PM

      • Excellent: that’s a good way to show the pattern in what is called iterative form, meaning that each line follows from what went before. When you say that the equation form above is better, I assume the reason you find it better is that it lets you skip ahead to any odd number as the smaller side and get the matching larger side and hypotenuse without having to work through all the previous sets of numbers.

        You can convert your iterative form to the direct kind by rewriting what you have and then generalizing the pattern:

        (2*1+1)*1+1 = 4
        (2*2+1)*2+2 = 12
        (2*3+1)*3+3 = 24
        (2*4+1)*4+4 = 40

        (2*n+1)*n+n = whatever

        For example, for a small side of 13, n = 6, and
        (2*6+1)*6+6 = 84,
        so the triangle has sides of 13, 84, 85.

        Steve Schwartzman

        July 27, 2015 at 9:26 PM

      • That is how I started, and worked it out the way Steve describes. To be fair, I have worked this kind of problem many many times, years ago.


        July 28, 2015 at 5:16 AM

        • I, too, have done many pattern generalizations over the years, but in this case it was the first time I’d come to a method of generating right triangles by generalizing kabeiser’s approach. The way I usually showed students was the one shoreacres hit upon below.

          Steve Schwartzman

          July 28, 2015 at 8:09 AM

          • Oh, shoreacres approach is far more elegant. Cool.


            July 29, 2015 at 7:02 PM

            • The way I usually phrased it to my students was: square the length of the shortest side, then split that square into two parts that are 1 apart from each other.

              Steve Schwartzman

              July 29, 2015 at 7:17 PM

  14. No, I didn’t notice the 63 in base 2. But, now I know that 63 in base 4 is 333.

    If I was that ant, I would stop walking forward.

    Jim in IA

    July 27, 2015 at 10:48 AM

  15. Well, I don’t know if this is right, but it seems to work.


    July 27, 2015 at 1:08 PM

    • Hmmm… not quite right. Back to the drawing board.


      July 27, 2015 at 1:46 PM

      • You’re right that it’s not right, because you’ve proposed a second-degree equation. An equation of this type can have only as many solutions as the degree, in this case two. Here the equation is true when x = 3 and also when x = -1 (which can’t be the side of a triangle).

        What you want isn’t an equation but a method: starting with an odd number x, do a certain thing to it, then some other thing, etc., such that two of those latter things produce the two sides that match up with the x to make the three sides of a right triangle.

        Above all, though, please don’t feel obliged to put a lot of time into this unless you really want to. I know you have a busy enough blogging life as it is, plus a regular life.

        Steve Schwartzman

        July 27, 2015 at 2:29 PM

        • Oh, I don’t feel obliged at all. Like writing haiku, it’s a fun way to pass the time at work.

          Besides, I think I have it now. Once I knew I was looking for a pattern, I wrote down the sets on a piece of paper,and messed around for a while. Once I squared the first numbers — 5, 7, 9, etc. — I saw it. Each pair of consecutive integers makes up the total of the first number squared. Five squared is 25, and the consecutive integers are 12 and 13 — a total of 25. Seven squared is 49, the total of 24 and 25 — and so on.

          It took a minute to figure out how to deal with less obvious larger numbers, but it seems that dividing by two will do it. For example, it would be 15- 112 -113. Fifteen squared is 225; divide, and — voila!


          July 27, 2015 at 3:11 PM

  16. Great capture. You lost me with your maths – not my strongest forte

    Raewyn's Photos

    July 27, 2015 at 1:42 PM

    • You’re with almost all the other commenters, which isn’t surprising, because people come here primarily for nature photographs, not math. I slip some math in on rare occasions, but most readers ignore it.

      Steve Schwartzman

      July 27, 2015 at 2:32 PM

  17. […] Three people have given solutions to the math problem posed in the recent post about the Aztec dancer: you can check out the comments there by Aggie, kabeiser, and shoreacres (in that […]

  18. Square the number, subtract 1, divide in half for one side, add back the 1 to that side for the 3rd side.

    Jean Wilson

    July 31, 2015 at 7:29 AM

    • Very good. That’s it indeed. If you look through the comments above you’ll find that Aggie, kabeiser, and shoreacres have also hit upon methods. Shoreacre’s is another way of saying what you’ve said, and the other two are different approaches.

      Steve Schwartzman

      July 31, 2015 at 7:56 AM

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